DISTRIBUTIONS OF QUADRATIC FUNCTIONALS OF THE FRACTIONAL BROWNIAN MOTION BASED ON A MARTINGALE APPROXIMATION
用鞅近似方法计算分数布朗运动二次泛函的分布，包括分数单位根分布，通过数值反演特征函数得到近似分布，并与模拟结果对比，发现近似分布的一个矩性质并猜想真实分布也有此性质。
The present paper deals with the distributions related to the fractional Brownian motion (fBm). In particular, we try to compute the distributions of (ratios of) its quadratic functionals, not by simulations, but by numerically inverting the associated characteristic functions (c.f.s). Among them is the fractional unit root distribution. It turns out that the derivation of the c.f.s based on the standard approaches used for the ordinary Bm is inapplicable. Here the martingale approximation to the fBm suggested in the literature is used to compute an approximation to the distributions of such functionals. The associated c.f. is obtained via the Fredholm determinant. Comparison of the first two moments of the approximate with true distributions is made, and simulations are conducted to examine the performance of the approximation. We also find an interesting moment property of the approximate fractional unit root distribution, and a conjecture is given that the same property will hold for the true fractional unit root distribution.